The Brazilian forward is understood to have had his release clause triggered with the Liverpool move.
Costa netted 10 goals in La Liga last season, and Anfield manager Brendan Rodgers aims to finalise a deal within the next few days.
"They have to be of the right quality, that's the bottom line," said Rodgers earlier.
"We've got money to spend on getting that quality in, but if it's not available I won't waste it for the sake of bringing a player in.
"We're really working hard to bring in the type of profile that can really help us push on."